Optimal. Leaf size=75 \[ \frac {e \sinh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)}-\frac {b c \log (F) \cosh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)} \]
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Rubi [A] time = 0.02, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {5475} \[ \frac {e \sinh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)}-\frac {b c \log (F) \cosh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)} \]
Antiderivative was successfully verified.
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Rule 5475
Rubi steps
\begin {align*} \int F^{c (a+b x)} \cosh (d+e x) \, dx &=-\frac {b c F^{c (a+b x)} \cosh (d+e x) \log (F)}{e^2-b^2 c^2 \log ^2(F)}+\frac {e F^{c (a+b x)} \sinh (d+e x)}{e^2-b^2 c^2 \log ^2(F)}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 50, normalized size = 0.67 \[ \frac {F^{c (a+b x)} (e \sinh (d+e x)-b c \log (F) \cosh (d+e x))}{(e-b c \log (F)) (b c \log (F)+e)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 246, normalized size = 3.28 \[ -\frac {{\left (e \cosh \left (e x + d\right )^{2} - {\left (b c \log \relax (F) - e\right )} \sinh \left (e x + d\right )^{2} - {\left (b c \cosh \left (e x + d\right )^{2} + b c\right )} \log \relax (F) - 2 \, {\left (b c \cosh \left (e x + d\right ) \log \relax (F) - e \cosh \left (e x + d\right )\right )} \sinh \left (e x + d\right ) - e\right )} \cosh \left ({\left (b c x + a c\right )} \log \relax (F)\right ) + {\left (e \cosh \left (e x + d\right )^{2} - {\left (b c \log \relax (F) - e\right )} \sinh \left (e x + d\right )^{2} - {\left (b c \cosh \left (e x + d\right )^{2} + b c\right )} \log \relax (F) - 2 \, {\left (b c \cosh \left (e x + d\right ) \log \relax (F) - e \cosh \left (e x + d\right )\right )} \sinh \left (e x + d\right ) - e\right )} \sinh \left ({\left (b c x + a c\right )} \log \relax (F)\right )}{2 \, {\left (b^{2} c^{2} \cosh \left (e x + d\right ) \log \relax (F)^{2} - e^{2} \cosh \left (e x + d\right ) + {\left (b^{2} c^{2} \log \relax (F)^{2} - e^{2}\right )} \sinh \left (e x + d\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.21, size = 611, normalized size = 8.15 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 74, normalized size = 0.99 \[ \frac {\left (\ln \relax (F ) b c \,{\mathrm e}^{2 e x +2 d}+b c \ln \relax (F )-e \,{\mathrm e}^{2 e x +2 d}+e \right ) {\mathrm e}^{-e x -d} F^{c \left (b x +a \right )}}{2 \left (b c \ln \relax (F )-e \right ) \left (e +b c \ln \relax (F )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 63, normalized size = 0.84 \[ \frac {F^{a c} e^{\left (b c x \log \relax (F) + e x + d\right )}}{2 \, {\left (b c \log \relax (F) + e\right )}} + \frac {F^{a c} e^{\left (b c x \log \relax (F) - e x\right )}}{2 \, {\left (b c e^{d} \log \relax (F) - e e^{d}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 74, normalized size = 0.99 \[ -\frac {F^{a\,c+b\,c\,x}\,{\mathrm {e}}^{-d-e\,x}\,\left (e-e\,{\mathrm {e}}^{2\,d+2\,e\,x}+b\,c\,\ln \relax (F)+b\,c\,{\mathrm {e}}^{2\,d+2\,e\,x}\,\ln \relax (F)\right )}{2\,\left (e^2-b^2\,c^2\,{\ln \relax (F)}^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.47, size = 316, normalized size = 4.21 \[ \begin {cases} - \frac {\left (-1\right )^{a c} \left (-1\right )^{- \frac {i e x}{\pi }} x \sinh {\left (d + e x \right )}}{2} + \frac {\left (-1\right )^{a c} \left (-1\right )^{- \frac {i e x}{\pi }} x \cosh {\left (d + e x \right )}}{2} + \frac {\left (-1\right )^{a c} \left (-1\right )^{- \frac {i e x}{\pi }} \sinh {\left (d + e x \right )}}{2 e} & \text {for}\: F = -1 \wedge b = - \frac {i e}{\pi c} \\x \cosh {\relax (d )} & \text {for}\: F = 1 \wedge e = 0 \\\tilde {\infty } e \left (e^{- \frac {e}{b c}}\right )^{a c} \left (e^{- \frac {e}{b c}}\right )^{b c x} \sinh {\left (d + e x \right )} + \tilde {\infty } e \left (e^{- \frac {e}{b c}}\right )^{a c} \left (e^{- \frac {e}{b c}}\right )^{b c x} \cosh {\left (d + e x \right )} & \text {for}\: F = e^{- \frac {e}{b c}} \\\tilde {\infty } e \left (e^{\frac {e}{b c}}\right )^{a c} \left (e^{\frac {e}{b c}}\right )^{b c x} \sinh {\left (d + e x \right )} + \tilde {\infty } e \left (e^{\frac {e}{b c}}\right )^{a c} \left (e^{\frac {e}{b c}}\right )^{b c x} \cosh {\left (d + e x \right )} & \text {for}\: F = e^{\frac {e}{b c}} \\\frac {F^{a c} F^{b c x} b c \log {\relax (F )} \cosh {\left (d + e x \right )}}{b^{2} c^{2} \log {\relax (F )}^{2} - e^{2}} - \frac {F^{a c} F^{b c x} e \sinh {\left (d + e x \right )}}{b^{2} c^{2} \log {\relax (F )}^{2} - e^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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